End Behavior of Functions
Objective
So what does it mean if a function is increasing?
So, the official definition is: “A function is increasing on an interval if ƒ(x1) < ƒ(x2) when x1 < x2 for any x-values x1 and x2 from the interval.”
Now, in plain English, it means that a function is increasing from one point to another, if the x’s are increasing and y’s are increasing at the same time on a given interval.
It’s a little hard to explain, so here’s an example:
Increasing function
Here’s the function:
As we can see, it looks like as:
x increases
y increases (or the graph goes up)
Now, we need an interval to tell whether or not this function can be called increasing or not, since the definition specifies on a given interval. �
So what does it mean if a function is decreasing?
So, the official definition is: “A function is decreasing on an interval if ƒ(x1) > ƒ(x2) when x1 < x2 for any x-values x1 and x2 from the interval.”
So basically this means that a function is decreasing from one point to another, if the x’s are increasing and y’s are decreasing at the same time on a given interval.
So it’s very similar to what we saw before, just backwards.
Decreasing function
We’re going to use the same function as we did before.
�
As we can see, it looks like as:
x increases
y decreases (or the graph goes down)
LOCAL MAXIMUM AND LOCAL MINIMUM OF A FUNCTION
A local maximum is a maximum that can be seen given a specific interval.
Like the graph we saw before, some graphs have what we call valleys (or low points) and peaks (or high points).
These highs and lows are considered the local maximums and local minimums of a function.
The peaks are the maximums, and the valleys are the minimum.
So why do we call them local?
Because they are specific to the interval given.
The actual maximum of a function is usually referred to as the global maximum, and is the highest point of the function at all intervals. �Or, in other words, the highest point of the entire function.
So let’s look at a quick example to tell the difference.
THE LOCAL MAXIMUM
Let’s use the same graph (because I’m lazy and don’t want to go making another one).
Again, it’s important to point out that the local maximum is not the highest point of the entire graph, it’s just the highest point of that interval.
THE LOCAL MINIMUM
Now let’s talk about the local minimum.
The local minimum is the same as the local maximum, except that it’s the lowest point of the graph.
So, take the same graph (again, lazy!).
The zeroes of a function
Finally, the zeroes of a function are where the graph crosses the x-axis.
This only occurs when you set y = 0.
They are also fairly easy to spot, so let’s look at an example:
As we can see from this graph, at (0,0) and at (3,0) it crosses/touches the x-axis.
So the zeroes of this graph would be: x = 0, x = 3.
And yes, that’s how you would write that.
Example:
As we can see, it seems like the graph is decreasing from -4 to -3, but then drastically increases from -3 on, so we would say this is locally increasing.
The local maximum is going to be at y = 9 since that’s the highest point of the graph, and the local minimum would be at y = 0, the lowest point of the graph.
The zero of the function (given the interval) would be at x = -3
AVERAGE RATE OF CHANGE
The average rate of change of a function is just that, the average change of the function between two intervals.
So, what does this mean?
Well, the official formula for finding the average rate of change is:
So what does this mean?
Well, we pick our interval, x1 and x2.
Then we find f(x1) and f(x2) (by plugging in our x values)
Then we plug everything into our equation.
EXAMPLE:
Find the average rate of change for the following function at the interval:
So, first we need to see what f(-4) actually is
To find it, we look at what the y value is at
x = -4
So, when x = -4, y = 1
Now we find f(0).
So, looking at the graph, we can see that:
When x = 0, y = 9
So f(0) = 9
Now we plug and chug!
2
So the average rate of change is 2!
End behavior
Last thing we’ll talk about concerning graphs (at least for right now) is their end behavior.
Basically, the end behavior is what the graph will do constantly in either the positive direction, or the negative direction.
The way we determine what the end behavior of a graph is, is by determining the direction we want to face, then determine what the graph does.
This is more complicated to explain than show, so here’s an example:
Example:
Determine the end behavior from the following function:
As we can see, as x increases, we can see that this graph seems
to fall to negative infinity.
So, to answer this question, we would say:
Likewise, if we want to look at as x decreases, we can say that
the graph seems to rise to positive infinity.
So, we would also say:
Example 2
Determine the end behavior from the following function:
As we can see, as x increases, we can see that this graph seems
to rise to positive infinity.
So, to answer this question, we would say:
Likewise, if we want to look at as x decreases, we can say that
the graph seems to rise to positive infinity still.
So, we would also say:
Last example:
Determine the end behavior from the following function:
As we can see, as x increases, we can see that this graph seems
to rise to positive infinity.
So, to answer this question, we would say:
Likewise, if we want to look at as x decreases, we can say that
the graph seems to fall to negative infinity.
So, we would also say: